Networks of Piecewise smooth Filippov systems and stability of synchronous periodic orbits

In this work, we considered networks of piecewise- smooth (PWS) differential equations of Filippov type, which we call PWS Filippov networks, whose single agent has an asymptotically stable periodic orbit, that becomes a synchronous periodic orbit for the network. We investigated the stability of the synchronous periodic orbit by using both the master stability function (MSF) tool and direct integration of the “regularized network”, that is the network obtained by replacing the PWS differential agents by a suitable smooth approximation. We present several new results on the class of Filippov PWS networks that depend on several coupling matrices. (i) We studied an associated MSF that depends on several coupling strengths, and (ii) we employed a one-parameter family of regularized networks, observed that it is equivalent to a network of regularized agents, and showed that its solutions converge to the solutions of the PWS network as the regularization parameter goes to 0. Furthermore, when the synchronous periodic orbit is asymptotically stable and does not slide on the discontinuity surface, (iii) we showed that the regularized network has an asymptotically stable synchronous periodic orbit as well. Finally, (iv) we performed detailed numerical experiments on two different problems, highlighting specific characteristics of PWS networks with a synchronous periodic orbit. More specifically, (a) first, we considered the case of a network of planar PWS oscillators, with the network depending on two different coupling parameters and the synchronous periodic trajectory undergoing sliding regime; then, (b) we considered a network where the single agent obeys 3D PWS dynamics with the synchronous periodic orbit undergoing repeated crossing of two distinct discontinuity planes. On our two problems, we also showed how the convergence behavior to the synchronous subspace can be used to obtain the same rates of attractivity predicted by the MSF.